Assessing the Validity of the Unireactant Lumped Approximation in Multireactant Systems Following First-Order Kinetics F. Xavier Malcata Escoia Superior de Biotecnologia, Universidade Catoiica Portuguesa, Rua Dr. Antonio Bernardino de Almeida, 4200 Porto, Portugal One of the major goals of applied chemistry is to design reactors able to carry out to a given extent a chemical reaction of interest employing the most favorable operating conditions from an economic point of view (I).Time constraints play a fundamental role in this particular, so the prior knowledge of the reaction(s) kinetics as obtained from studies of pure chemistry is required. Enhancements of the reaction rate may be brought about by either a temperature increase or the use of a catalyst. Catalysts are able to speed up reactions because they decrease the activation energy of one or more rate-controlling steps pertaining to the mechanism of the reaction, and in this respect they act much more selectively than a plain temperature increase. Catalysts can he of synthetic origin (inorganic catalysts) or he obtained from biological sources (enzymes). Most biological catalysts possess a very high degree of specificity due to their role in vivo. Some enzymes show, however, large affinity to a wide variety of polymeric substrates provided that these suhstrates share a common type of labile covalent bond (2). In the latter situation, the various reactants compete with each other for the active site of the enzyme regardless of their sequence of monomer residues or overall molecular weieht. Examples documented in the literature include the action of such hydrolases as lysozyme on muropolysacon charides of harterial cell walls (3) . and amvloalucosidase - amylose (4,5). Of particular interest here are the reactions effected by soiubleexo-hydrolases (i.e..enzymer that cleaveester,glycosidic, or peptide bonds nexr to the ends of polymeric carbon backbon&;thus releasing monomeric subunits) on complex aqueous mixtures of suhstrates consisting of linear biopolymers of various chain lengths. The enzyme kinetics is assdmed to be accurately described by a multisubstrate Michaelis-Menten rate equation (2) with large dissociation constant for the enzyme-substrate complex contributed by every substrate (i.e., KM.; >> Ci). The general stoichiometry can be represented as follows ~
where Si denotes a substrate made of i monomeric subunits. The pseudo-first-order rate constant is defined ask; = umaX,j/
KM,;. 288
Journal of Chemical Education
The chemical reaction is assumed to be carried out in a continuously stirred tank reactor (CSTR) under isothermal conditions and absence of enzyme deactivation. In this type .. of reartor the ease of operation, low construction costs, and absence of concentration gradients of the ytirred hatch counterpart are coupled with the possibility of steady-state operation and lack of disturbance on the reacting fluid upon sampling that are characteristic of continuous-flow reactors (6). A state of perfect micromixing within the reacting fluid is assumed throughout (7). A mass balance to every type of reactant andlor product molecule then reads
where the first term (or the only term) in the LHS denotes the inlet molar flow rate, the second term (or the remaining terms) in the LHS represent the molar rate of production, the first term (or the only term) in the RHS accounts for the outlet molar flow rate, and the second term in the RHS arises from the molar rate of consum~tion. Solving the first of eqs 2 with respect to CNand combining the above eauations in arecursive fashion from i = 1down to i = N - 1,one obtains i" c,=- 1 +C ,Do,
Frequently in experiments and process operation, it is neither possible nor practical to measure and monitor the concentrations of all species present in the reaction mixture. One way to circumvent this difficulty is by lumping together all species containing any number of monomeric subunits greater than unity into a single imaginary species. Such hypothetical species can he viewed as a pseudo-monomer that, upon enzymatic action, releases a true monomer and some other inert moiety, according to the following chemical equation
Lumping of this type is very widespread in all branches of chemical kinetics. Although not often explicitly recognized in the literature of applied chemistry, this kind of lumping procedure is a pervasive practice when investigating, modeling, or designing chemical reactors (2). Possible shortcomings of the unireactant implicit assumption can be easily made apparent for the case of reactions following approximate or true first-order kinetics performed in a CSTR by setting kl = kn = . . . = k~ = ko and C1.i"= C2.e - . . . = CN,~" = COin the above multireactant model. In this case, eq 3 becomes
If all the reactants are lumped into a single, hypothetical reactant, then the mass balances to this pseudo-monomer and to the true monomer released by chemical reaction may be written as QC;,$,= QC;
+ VkoC;
Straightforward manipulation of eq 8 yields
In the case of having all true reactants share the same concentration Co then the corresponding situation involving the lumped reactant approach leads to
where advantage was taken from the principle of finite induction (9). Equation 9 then takes the form N + 1)- 2)C0 c; = ( N (2(1+ Da,)
Defining a ratio R of the final concentration of monomer as predicted by the multireactant model to the final concentration of monomer as predicted by the lumped, unireactant model, one gets after some algebraic manipulation
N t 2
Expansion of the multiple summations in the foregoing expressions according to the properties of geometric series (8)yields
Expansion of the geometrical series left in the last expression above gives the following simplified form
Plots of the ratio R covering a range of values of N with physical interest are depicted in the figure. For smallN both
Plots 01 Ibratio of the predicted outlet concentration of monomer using the muttireactantapproxlmatlonto the predicted outlet concenhation of monomer usingthe iwnped, unireectantapproximation,R, v e n w theDamkDhier number associated With the react- operating conditions end the kinetic propenies, DB,for various aders of magnitude of the number of monomer moieties In the largest biapolymer. N.
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approache$ yield approximately t h e same result in the whole range of Da,a s expected. For very low and very high values of t h e Damkohler number there is n o significant difference in the results obtained using either approach, provided t h a t the number of monomer subunits contributing to the largest biopolymer does not become excessively high. This means t h a t the conclusions are model-independent if t h e time scale associated with t h e reaction kinetics is either much smaller or much larger than t h e average residence time of reacting fluid in t h e reactor. It can b e also observed t h a t the ratio R is less than unity for t h e complete range of Damkohler numbers. Therefore, designing a CSTR for the production of a given monomer under the assumptions listed ahove for t h e lumped approach will always result in actual conversion extents lower t h a n desired. This result was somehow qualitatively anticipated because the exponential variation of reactant concentration with space time in a CSTR does not linearly scale down with t h e reactant concentration: the l u m ~ e da ~ ~ r o a ctakes h advantage of the high concentrations of s i b s t r a t e which produci very high reaction rates, whereas t h e multireactant approach considers all substrates a t lower concentration levels which produce much lower overall reaction rates. It is also interesting t o note t h a t for large N, t h e values of R are well described by the following asymptotic expression Lim R = N-=
2(1 +Do,)
N
Hence, t h e relative error associated with the use of the lumpe d approach instead of the multireactant approach becomes a linear increasing function of N in a log-log scale for each Da.This limiting behavior can be used a s a n approximation for linear biopolymers of carhohydrate residues: for N larger than, say, 100, these can be accurately handled a s infinitely long reactant molecules. If a change in ki exists from one reactant t o another, or if t h e initial concentrations of t h e reactants are not equal, then the differences between t h e two approaches hecome more important. T h e ahove derivation serves the educational purpose of providing a quantitative perspective of the differences in chemical reactor design arising solely from the sim-
plification of assuming a lumped reactant instead of considering all reactants a s acting independently.
Nomenclature C; = eoneentration of substrate containing i monomer residues Cp =concentration of lumped substrate containing one single labile bond Da; = k;VIQ, Damk6hler number I' = inert moiety pseudo-first-order kinetic constant k; = umax,ilK~,i, IfM,; = Michaclis-Menten constant for the substrate containing i monomer residues N = number of monomer residues of largest biopolymer reactant Q = volumetric flow rate of liquid through the reactor R = CI/CIa,ratio of predicted monomer eoneentration using the multireactant aooroacb to the re dieted monomer concen.. tration using the lumped, unireactant approach S, = substrate containing L monomer residues S* = lumped substrate urn.., = reaction rate under saturating conditions of the enzyme active site with substrate V = volume of reactor Subscripts i = dummy integer variable in = referring to the inlet conditions j = dummy integer variable k = dummy integer variable 1 = dummy integer variable o = reference constant value Superscripts * = using the lumped unireactant approach ' = using the multireactant approach under the assumptions that k, = ka = = k N = ko and C,,:, = C2,in= . . . = CN,~,, = CO
...
Literature Cited 1. Levenspiel. 0.Chemical Reaction Engineering; Wiley: New Yark, 1972; pp 30-34. 2. Bailey, J. E.: Ollis, D.F. Biochamirol Enginr~ringFundomsntals: McCraw-Hill: New York. 1986: pp89.116-120. 3. Chipman. D. M. Biochemistry 1971,10,1714. 4. Hiromi, K.;Ogaus,K.: Nakanishi,N.:Ono,S. J.Biochem. 1966.60.439. 5 . 0no.S.:Hir0rni.K.; 2inbo.M. J.Bioch~m.1964.55. 315. 6. Hill. C. G. An Inrrodurtion to Chemicoi Engineering Kineties and Reactor Design; Wiley: New York. 197tp249. 7. Zwicfering,Th. N. ChemEng. Sci. 1959,9,1.
8. Stephensun.M.MalhemolVolMelhods/orS~iiiiiWud~nl~:Longman:London.1973:
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